The area (in square units) of the region $A = \{(x, y) :|x|+|y| ≤1, 2y^2 ≥|x|\}$ is

$\frac{5}{6}$

The area S of the shaded region is given by

$S=4\int\limits_{0}^{1/2}(y_2-y_1)dx=4\int\limits_{0}^{1/2}\left((1-x)-\sqrt{\frac{x}{2}}\right)dx$

$⇒S=4\left[x-\frac{x^2}{2}-\frac{2}{3\sqrt{2}}x^{3/2}\right]_{0}^{1/2}=4\left[\frac{1}{2}-\frac{1}{8}-\frac{\sqrt{2}}{3}×\frac{1}{2\sqrt{2}}\right]$

$⇒S=4×\frac{5}{24}=\frac{5}{6}$